Let there be a polynomial whose roots to be found and let us start with initial guesses $x_0\ x_1\ x_2$. Let the quadratic $a(x-x_2)^2+b(x-x_2)+c$ developed from these values have complex roots. Denote them as $\alpha+\beta i $ and $\alpha-\beta i$. One of them is to be chosen as $x_3$, the first estimate. Which root should I proceed with? It is said that one should choose the largest denominator from $x_2+\frac{-2c}{b\pm \sqrt {b^2-4ac}} $ to determine $x_3$ but mathematically I cannot compare two complex numbers and choose one over the other.
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2You compare the magnitude of the denominators to determine which is largest. And if you get equal magnitudes then it usually does not matter which is taken. – Simply Beautiful Art Dec 31 '20 at 14:27
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@SimplyBeautifulArt If $b^2-4ac<0$ I get a complex number there. Are you referring to the modulus of it? – Ali Kıral Dec 31 '20 at 14:37
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1Yes, you compare the moduli of both denominators, even in the real case. – Simply Beautiful Art Dec 31 '20 at 23:17